Nearly Space-Optimal Graph and Hypergraph Sparsification in Insertion-Only Data Streams

TL;DR

This paper introduces nearly space-optimal algorithms for graph and hypergraph sparsification in insertion-only data streams, achieving approximation guarantees close to offline algorithms with significantly reduced space complexity.

Contribution

It presents new streaming algorithms for hypergraph and graph sparsification that match offline sample complexity up to poly-logarithmic factors, and introduces space-efficient algorithms for min-cut and adversarial robustness.

Findings

Achieves $(1+\varepsilon)$-approximation for hypergraph sparsification with near-optimal space.

Improves space bounds for graph sparsification by a factor of $\log n$.

Provides space-efficient algorithms for min-cut approximation and adversarially robust hypergraph sparsification.

Abstract

We study the problem of graph and hypergraph sparsification in insertion-only data streams. The input is a hypergraph $H=(V, E, w)$ with $n$ nodes, $m$ hyperedges, and rank $r$, and the goal is to compute a hypergraph $\widehat{H}$ that preserves the energy of each vector $x \in \mathbb{R}^n$ in $H$, up to a small multiplicative error. In this paper, we give a streaming algorithm that achieves a $(1+\varepsilon)$-approximation, using $\frac{rn}{\varepsilon^2} \log^2 n \log r \cdot\text{poly}(\log \log m)$ bits of space, matching the sample complexity of the best known offline algorithm up to $\text{poly}(\log \log m)$ factors. Our approach also provides a streaming algorithm for graph sparsification that achieves a $(1+\varepsilon)$-approximation, using $\frac{n}{\varepsilon^2} \log n \cdot\text{poly}(\log\log n)$ bits of space, improving the current bound by $\log n$ factors. Furthermore, we give a space-efficient streaming algorithm for min-cut approximation. Along the way, we present an online algorithm for $(1+\varepsilon)$-hypergraph sparsification, which is optimal up to poly-logarithmic factors. As a result, we achieve $(1+\varepsilon)$-hypergraph sparsification in the sliding window model, with space optimal up to poly-logarithmic factors. Lastly, we give an adversarially robust algorithm for hypergraph sparsification using $\frac{n}{\varepsilon^2} \cdot\text{poly}(r, \log n, \log r, \log \log m)$ bits of space.